A continuous variable whose effects you want to control for (partial out) regarding the DV before looking at group differences on that DV

The covariate has a linear relationship with the DV

3Mancova

Testing for significantly different groups is conceptually the same as in Anova.

We compare an estimate of the variance between groups to that which comes from within groups

The difference is that we adjust the dependent variable by removing the variation that can be explained by the covariate

What is left over is the adjusted DV, Yadj

4Mancova

In Mancova the linear combination of DVs is adjusted for one or more covariates.

The adjusted linear combinations of the DVs is the combination that would have been had all of the subjects scored the same on the covariate.

For the significance tests, wed go about them the same way as with regular Manova

5Caveats to Mancova

One should think hard before utilizing Ancova/Mancova

1. You need a theoretical reason for considering something to be a covariate, and want to have a minimum number of covariates

2. Many recommend against using covariates with intact groups (e.g. different schools)

This is because the adjustment still does not mean that the two groups are equivalent on that covariate in reality

Essentially you are asking about what the data would look like if it werent the data it is

Since we are dealing with intact groups, they may still differ on any number of characteristics which may cause unequal group means on y that are confounded with any treatment effects.

3. If one cannot imagine groups being equivalent on the covariate in the real world, then it doesnt make sense to use a procedure that equates them on that variable

Weight as a cv with gender an IV

6Caveats to Mancova

4. Additional assumptions must be met for Ancova designs (homogeneity of regression)

5. The covariate should be independent of the IVs

E.g. the covariate cannot be measured after a treatment effect has been applied, as it would mean that part of whats being partialled out of the DV may actually be due to the treatment

These all go with other issues/problems regarding Anova approaches in general

Also note that other approaches may be available, particularly with pre post settings

Analyses on gain scores

Mixed Design

Solomon 4-group

7Testing interactions

A factorial MANOVA may be used to determine whether or not two or more categorical independent variables (and their interactions) significantly affect optimally weighted linear combinations of two or more normally distributed dependent variables.

As in univariate factorial ANOVA, we will first take note of the interaction, as it will inform us whether to qualify any main effects or not

8Interactions

One could inspect the univariate interactions for each DV, but this does not really get at the heart of the matter with its disregard of DV correlations

We could also, in a similar fashion as we suggested in the one way design as a post hoc, perform multivariate one-way designs for each level of the second variable

9Interactions

However there is a problem this time around with such an approach with interactions

With the interaction setting, neither set of weights would likely be the same as that which maximized the multivariate interaction

10Interactions

A significant multivariate interaction means that the effect of one IV depends upon the levels of another IV, where the multivariate DV is one or more canonical variates

Compute each subjects canonical variate scores for each root that is significant and then do simple effects analysis on the corresponding canonical variate.

i.e. Univariate anovas on the variate scores

Note this would be an option in the one-way MANOVA scenario as well as we have discussed

Now run simple effects tests, i.e. univariate anovas on the single DV of variate scores

In SPSS this involves /EMMEANS COMPARE

Look for differences of method at levels of high or low achievement or vice versa.

16OutputHigh AchieversLow Achievers 17Simple Effects

Recall that simple effects are not just a breakdown of an interaction, but the variance attributable to the interaction and the main effect being studied

If A at levels of B the variance in the simple effects come from variance of A and AB

While your interaction tells you the simple effects are different, the simple effects tell are telling you just whether the mean differences for that grouping variable are significantly different from zero (just like a one-way ANOVA)

18Graphically

Sig difference of method for high achievers but not for low achievers

One could then perform more post hocs or go by a descriptive account

Methods 4 seems to be best and 1 worst for high achievers but low achievers arent really showing any differences

19Causality in Manova

Path analytic approach to Manova

20

Consider a 2 group 3 DV set up as here

T the dummy coded grouping variable

Ys are the DVs

Arrows are paths

Lower case letters coefficients representing the strength of that path

21

U, V, and W are disturbance terms i.e. error

Source of variability in the DVs not accounted for by the model

Example, U represents all causes of Y1 not caused by the treatment assignment

The disturbances of the DVs are correlated with one another, though precisely how (causally) is not specified

Hence double arrows

22

The omnibus Manova null hype is that the treatment group means are the same on the DVs

In terms of the path analysis, we are saying the treatment has zero correlation with the DVs

Coefficients 0

The test can be seen as a comparison of a (full) model such as the figure vs. one (reduced) in which there are no causal paths extending from T

23

A follow up of univariate tests can be seen here as well

The omnibus rejected the idea that the coefficients 0

The univariate approach would be testing whether each particular coefficient 0

24

Here is a case involving 4 treatment groups

K-1 3 dummy variables

The same approach applies, i.e. the omnibus test of whether any coefficients a-i are significantly different from zero

A simultaneous test of a,b, and c would be equivalent to the unvariate test on Y1

25Step-down test

This diagram represents a two group IV with 3 DVs

The equations are indicative of the regression approach, but the analysis can also be thought of as an Ancova with any prior DVs as the covariate(s)

Note that unlike previously, this suggests a causal ordering to the DVs rather than a simple correlation

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